Single-variable delay-differential equation approximations of the Fitzhugh-Nagumo and Hodgkin-Huxley models

• Published in: Communications in nonlinear science & numerical simulation Vol. 82; p. 105066
• Main Authors: Rameh, Raffael Bechara, Cherry, Elizabeth M., dos Santos, Rodrigo Weber
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 01.03.2020; Elsevier Science Ltd
• Subjects: Approximation; Biodiversity; Cellular biology; Delay; Delay-differential equation; Differential equations; Electrophysiology; Fitzhugh-Nagumo; Hodgkin-Huxley; Mathematical models; Hodgkin-Huxley; Fitzhugh-Nagumo; Delay-differential equation
• ISSN: 1007-5704; 1878-7274
• DOI: 10.1016/j.cnsns.2019.105066

•Traditional models of neural action potentials use systems of differential equations.•Effects of secondary variables are similar to delays applied to membrane potential.•New excitable models based on single-variable delay-differential equation are presented.•Single-variable models with delays preserve important properties of action potentials. In this work, we revisit two traditional models of action potential generation in excitable cells, the Hodgkin-Huxley (HH) and the FitzHugh-Nagumo (FHN) models. The main goal is to evaluate the possibility of modeling the generation of the action potential via a single delay-differential equation (DDE). Delay-differential equations are important mathematical tools and can reproduce a great diversity of biological phenomena. However, their use in modeling of action potentials is still incipient. In this paper, we present new models based on single delay-differential equation. The solutions of the new models are similar to those of the original HH and FHN models. Based on these results, we claim that delay-differential equations can also be used as building blocks in the development of models of cell electrophysiology.